360 
Proceedings of the Royal Society of Edinburgh. 
[Sess. 
Put 
-Ltani=cos 2y x 
and 
J tan ^ = cos 2y{ ; 
then 
l Y = tan 2 y[ , 
giving 
2i = + e ^ c - 
and 
1 i 1 
v i = — log e — . 
7T q 1 
Then 
w z = 
and 
2 
— - n-,w, 
7 r 
13 
W 
If we put f 3 = 180° — l v we can interchange the suffixes x and 3 so that all 
the processes are available in either case. 
YI. 
The two hypergeometrical series of $ 4 are not sensitive ; and their 
tabulation by Mr Frank Robbins is given in the Appendix. The periods 
are sensitive, but in the present method of calculation almost the whole of 
the variation is thrown on the simple factors 1/cos 2 h and 1/sin 2 1 ± . 
A A 
If we remember that cos = sin h ; the following argument, here limited 
to sin can be extended. When i x is under 90°, and an accuracy of two 
units in the seventh place of the decimals of the logarithms of the hyper- 
geometrical series will suffice, the use of series can be avoided by introducing 
the following approximate formula : — 
(~M’ 2 ’ sin A) =log h- 
log F( - _ 
— sin 2 A 
72 2 
1 - 4 sin 2 -R u 
/ ° N 100 
" [6 ' lo,,I (roo)’ 
and 
log F( T ’ T ’ 2 ’ sin2 ^ “ log 1 + 
5 - o fc-i 
72 Sm i 
U'- ip 
[ + [ 6« 4 »](i)lf 
